Brown's position was under threat after Hull's poor start to their second season in the top flight but Sunday's victory, their third in 12 games, bought him some time.
Matthew Etherington put Stoke ahead in the 29th minute with Seyi Olofinjana curling in a neat equaliser after 62 before Venegoor cashed in on a blunder by Stoke keeper Thomas Sorensen.
Everton won 2-1 at West Ham United to end a run of seven games without a win in all competitions.
They showed little on Sunday until Louis Saha shot them ahead after 27 minutes.
Dan Gosling doubled the lead after 63 minutes and though Tony Hibbert put the ball in his own net a minute later as he attempted to clear Junior Stanislas's goal-bound effort Everton held on to take the points.
Wigan Athletic drew 1-1 at home with Fulham while leaders Chelsea were playing Manchester United at Stamford Bridge (4 p.m. British time).
(Editing by Clare Fallon)